Дисертації з теми "Fixed-point equation"
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Dunn, Kyle George. "An Integral Equation Method for Solving Second-Order Viscoelastic Cell Motility Models." Digital WPI, 2014. https://digitalcommons.wpi.edu/etd-theses/578.
Повний текст джерелаKang, Jinghong. "The Computational Kleinman-Newton Method in Solving Nonlinear Nonquadratic Control Problems." Diss., Virginia Tech, 1998. http://hdl.handle.net/10919/30435.
Повний текст джерелаPh. D.
Ertem, Turker. "Asymptotic Integration Of Dynamical Systems." Phd thesis, METU, 2013. http://etd.lib.metu.edu.tr/upload/12615405/index.pdf.
Повний текст джерела&prime
= f (t, x) (0.1) and the solutions 1 and t of x&prime
&prime
= 0. More specifically, the existence of a solution of (0.1) asymptotic to x(t) = at + b, a, b &isin
R has been obtained. In this thesis we investigate in a systematic way the asymptotic behavior as t &rarr
&infin
of solutions of a class of differential equations of the form (p(t)x&prime
)&prime
+ q(t)x = f (t, x), t &ge
t_0 (0.2) and (p(t)x&prime
)&prime
+ q(t)x = g(t, x, x&prime
), t &ge
t_0 (0.3) by the help of principal u(t) and nonprincipal v(t) solutions of the corresponding homogeneous equation (p(t)x&prime
)&prime
+ q(t)x = 0, t &ge
t_0. (0.4) Here, t_0 &ge
0 is a real number, p &isin
C([t_0,&infin
), (0,&infin
)), q &isin
C([t_0,&infin
),R), f &isin
C([t_0,&infin
) ×
R,R) and g &isin
C([t0,&infin
) ×
R ×
R,R). Our argument is based on the idea of writing the solution of x&prime
&prime
= 0 in terms of principal and nonprincipal solutions as x(t) = av(t) + bu(t), where v(t) = t and u(t) = 1. In the proofs, Banach and Schauder&rsquo
s fixed point theorems are used. The compactness of the operator is obtained by employing the compactness criteria of Riesz and Avramescu. The thesis consists of three chapters. Chapter 1 is introductory and provides statement of the problem, literature review, and basic definitions and theorems. In Chapter 2 first we deal with some asymptotic relationships between the solutions of (0.2) and the principal u(t) and nonprincipal v(t) solutions of (0.4). Then we present existence of a monotone positive solution of (0.3) with prescribed asimptotic behavior. In Chapter 3 we introduce the existence of solution of a singular boundary value problem to the Equation (0.2).
Čambor, Michal. "Paralelní řešení parciálních diferenciálnich rovnic." Master's thesis, Vysoké učení technické v Brně. Fakulta informačních technologií, 2011. http://www.nusl.cz/ntk/nusl-412855.
Повний текст джерелаRocha, Suelen de Souza. "Soluções clássicas para uma equação elíptica semilinear não homogênea." Universidade Federal da Paraíba, 2011. http://tede.biblioteca.ufpb.br:8080/handle/tede/8051.
Повний текст джерелаMade available in DSpace on 2016-03-29T13:33:49Z (GMT). No. of bitstreams: 1 arquivo total.pdf: 5320246 bytes, checksum: 158dd460a20ce46c96d4a34623612264 (MD5) Previous issue date: 2011-08-25
This work is mainly concerned with the existence and nonexistence of classical solution to the nonhomogeneous semilinear equation Δu + up + f(x) = 0 in Rn, u > 0 in Rn, when n 3, where f 0 is a Hölder continuous function. The nonexistence of classical solution is established when 1 < p n=(n 2). For p > n=(n 2) there may be both existence and nonexistence results depending on the asymptotic behavior of f at infinity. The existence results were obtained by employed sub and supersolutions techniques and fixed point theorem. For the nonexistence of classical solution we used a priori integral estimates obtained via averaging.
Neste trabalho, estamos interessados na existência e não existência de solução clássica para a equação não homogênea semilinear Δu + up + f(x) = 0 em Rn; u > 0 em Rn, n 3 onde f 0 é uma função Hölder contínua. A não existência de solução clássica é estabelecida quando 1 < p n=(n 2). Para p > n=(n 2), temos resultados de existência e não existência de solução clássica, dependendo do comportamento assin- tótico de f no infinito. Os resultados de existência foram obtidos usando o método de sub e supersolução e teoremas de ponto fixo. A não existência de solução clássica é obtida usando-se estimativas integrais a priori via média esférica.
Rizzolo, Douglas. "Approximating Solutions to Differential Equations via Fixed Point Theory." Scholarship @ Claremont, 2008. https://scholarship.claremont.edu/hmc_theses/213.
Повний текст джерелаSun, Xun. "Twin solutions of even order boundary value problems for ordinary differential equations and finite difference equations." [Huntington, WV : Marshall University Libraries], 2009. http://www.marshall.edu/etd/descript.asp?ref=1014.
Повний текст джерелаMentemeier, Sebastian [Verfasser], and Gerold [Akademischer Betreuer] Alsmeyer. "On multivariate stochastic fixed point equations / Sebastian Mentemeier. Betreuer: Gerold Alsmeyer." Münster : Universitäts- und Landesbibliothek der Westfälischen Wilhelms-Universität, 2013. http://d-nb.info/1031885455/34.
Повний текст джерелаCremins, Casey Timothy. "Fixed point indices and existence theorems for semilinear equations in cones." Thesis, University of Glasgow, 1997. http://theses.gla.ac.uk/3520/.
Повний текст джерелаTiwari, Abhishek. "ANALYTICAL METHODS FOR TRANSPORT EQUATIONS IN SIMILARITY FORM." UKnowledge, 2007. http://uknowledge.uky.edu/gradschool_theses/457.
Повний текст джерелаDyrssen, Hannah. "Valuation and Optimal Strategies in Markets Experiencing Shocks." Doctoral thesis, Uppsala universitet, Tillämpad matematik och statistik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-316578.
Повний текст джерелаNdao, Mamadou. "Estimation de la vitesse de retour à l'équilibre dans les équations de Fokker-Planck." Thesis, Université Paris-Saclay (ComUE), 2018. http://www.theses.fr/2018SACLV036/document.
Повний текст джерелаThis thesis is devoted to the Fokker-Planck équation partial_t f =∆f + div(E f).It is divided into two parts. The rst part deals with the linear problem. In this part we consider a vector E(x) depending only on x. It is composed of chapters 3, 4 and 5. In chapter 3 we prove that the linear operator Lf :=∆f + div(Ef ) is an in nitesimal generator of a strong continuous semigroup (SL(t))_{t≥0}. We establish also that (SL(t))_{t≥0} is positive and ultracontractive. In chapter 4 we show how an adequate decomposition of the linear operator L allows us to deduce interesting properties for the semigroup (SL(t))_{t≥0}. Indeed using this decomposition we prove that (SL(t))_{t≥0} is a bounded semigroup. In the last chapter of this part we establish that the linear Fokker-Planck admits a unique steady state. Moreover this stationary solution is asymptotically stable.In the nonlinear part we consider a vector eld of the form E(x, f ) := x +nabla (a *f ), where a and f are regular functions. It is composed of two chapters. In chapter 6 we establish that fora in W^{2,infini}_locthe nonlinear problem has a unique local solution in L^2_{K_alpha}(R^d); . To end this part we prove in chapter 7 that the nonlinear problem has a unique global solution in L^2_k(R^d). This solution depends continuously on the data
Khalid, Adeel S. "Development and Implementation of Rotorcraft Preliminary Design Methodology using Multidisciplinary Design Optimization." Diss., Georgia Institute of Technology, 2006. http://hdl.handle.net/1853/14013.
Повний текст джерелаVážanová, Gabriela. "Existence a vlastnosti globálních řešení funkcionálních diferenciálních rovnic smíšeného typu." Doctoral thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2020. http://www.nusl.cz/ntk/nusl-433560.
Повний текст джерелаHopkins, Britney Henderson Johnny. "Multiplicity of positive solutions of even-order nonhomogeneous boundary value problems." Waco, Tex. : Baylor University, 2009. http://hdl.handle.net/2104/5323.
Повний текст джерелаMu, Tingshu. "Backward stochastic differential equations and applications : optimal switching, stochastic games, partial differential equations and mean-field." Thesis, Le Mans, 2020. http://www.theses.fr/2020LEMA1023.
Повний текст джерелаThis thesis is related to Doubly Reflected Backward Stochastic Differential Equations (DRBSDEs) with two obstacles and their applications in zero-sum stochastic switching games, systems of partial differential equations, mean-field problems.There are two parts in this thesis. The first part deals with optimal stochastic switching and is composed of two works. In the first work we prove the existence of the solution of a system of DRBSDEs with bilateral interconnected obstacles in a probabilistic framework. This problem is related to a zero-sum switching game. Then we tackle the problem of the uniqueness of the solution. Finally, we apply the obtained results and prove that, without the usual monotonicity condition, the associated PDE system has a unique solution in viscosity sense. In the second work, we also consider a system of DRBSDEs with bilateral interconnected obstacles in the markovian framework. The difference between this work and the first one lies in the fact that switching does not work in the same way. In this second framework, when switching is operated, the system is put in the following state regardless of which player decides to switch. This difference is fundamental and largely complicates the problem of the existence of the solution of the system. Nevertheless, in the Markovian framework we show this existence and give a uniqueness result by the Perron’s method. Later on, two particular switching games are analyzed.In the second part we study a one-dimensional Reflected BSDE with two obstacles of mean-field type. By the fixed point method, we show the existence and uniqueness of the solution in connection with the integrality of the data
LIMA, Annaxsuel Araújo de. "O método de sub e supersolução e aplicações a problemas elípticos." Universidade Federal de Campina Grande, 2011. http://dspace.sti.ufcg.edu.br:8080/jspui/handle/riufcg/1241.
Повний текст джерелаMade available in DSpace on 2018-07-25T17:20:25Z (GMT). No. of bitstreams: 1 ANNAXSUEL ARAÚJO DE LIMA - DISSERTAÇÃO PPGMAT 2011..pdf: 581866 bytes, checksum: cc44cd422d4a48ddad0354f215805918 (MD5) Previous issue date: 2011-04
Neste trabalho, apresentamos métodos envolvendo sub e supersolução para estudar a existência de solução de certas equações elípticas.
In this work, we present methods involving sub and supersolution to study the existence of solution of certain elliptic equations.
Marques, Dayvid Geverson Lopes. "Um Teorema de Ponto Fixo e Aplicações a Equações Elípticas Semilineares." Universidade Federal da Paraíba, 2012. http://tede.biblioteca.ufpb.br:8080/handle/tede/7370.
Повний текст джерелаCoordenação de Aperfeiçoamento de Pessoal de Nível Superior
In this work, we study a fixed point theorem for increasing operators in ordered normed spaces and we apply it in order to obtain results of existence of weak solution for semilinear elliptic equations of type 8<: ---u = f(x; u) + h; in u = 0; on @ ; where - RN is a smooth domain, f : -R --! R satisfies some convenient conditions and h 2 H--1(.
Neste trabalho, estudamos um teorema de ponto fixo para operadores crescentes em espaços vetoriais ordenados e o aplicamos para obter resultados de existência de solução fraca para problemas elípticos semilineares do tipo 8<: ---u = f(x; u) + h; em u = 0; sobre @ em que - RN é um domínio suave, f : - R ! R satisfaz algumas condições convenientes e h 2 H- -1(:).
Fernandes, Jairo. "Equações de diferenças e aplicações." reponame:Repositório Institucional da UFABC, 2016.
Знайти повний текст джерелаDissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2016.
Apresentamos neste trabalho um estudo sobre as equações de diferenças autônomas li-neares e não lineares que descreve um sistema dinâmico discreto. Para o caso linear, o objeti-vo foi encontrar uma solução analítica da evolução temporal do sistema e a partir desta solu-ção estudamos a estabilidade do sistema. No caso do não linear, na impossibilidade de deter-minar uma solução analítica, o que procuramos foi uma compreensão sobre a evolução quali-tativa do sistema, ou seja, fizemos um estudo qualitativo de uma família de mapas logísticos discretos, onde a partir da variação de um parâmetro verificamos alguns comportamentos co-mo: pontos fixos, órbitas periódicas, bifurcação e caos. Em ambos os casos, estudamos alguns modelos simples relacionados à Economia, Demografia ou Ecologia como exemplos de apli-cações dos aspectos teóricos estudados.
Here we present a study of the equations of linear and nonlinear autonomous differ-ences that describes a dynamic discrete system. For the linear case, the objective was to find an analytical solution of the time evolution of the system and from this solution we study the system stability. In the case of nonlinear, it is impossible to determine an analytical solution, what we seek is an understanding of the qualitative evolution of the system, ie, did a qualita-tive study of a family of discrete logistic maps, where from the change in a parameter we found some behaviors such as fixed points, periodic orbits, bifurcation and chaos. In both cas-es, we study some simple models related to Economics, Demography and Ecology as exam-ples of applications of the theoretical aspects studied.
SILVA, Geizane Lima da. "Soluções blow-up para uma classe de equações elípticas." Universidade Federal de Campina Grande, 2010. http://dspace.sti.ufcg.edu.br:8080/jspui/handle/riufcg/1231.
Повний текст джерелаMade available in DSpace on 2018-07-24T16:01:03Z (GMT). No. of bitstreams: 1 GEIZANE LIMA DA SILVA - DISSERTAÇÃO PPGMAT 2010..pdf: 596736 bytes, checksum: d02e34d40e7147e46c734ba297c181bf (MD5) Previous issue date: 2010-03
Capes
Neste trabalho estudamos a existência de soluções positivas do tipo blow-up para uma classe de equações elípticas semilineares. Usamos argumentos desenvolvidos por Cîrstea & Radulescu [6], Lair & Wood [20] e as técnicas empregadas são o Método de Sub e Supersolução, Teoremas de Ponto Fixo e em alguns resultados exploramos a simetria radial e algumas estimativas para equações elípticas.
In this work we studied the existence of blow-up positive solutions for the class of semilinear elliptic equations. We used arguments developed by Cîrstea & Radulescu [6], and by Lair & Shaker [20] and the techniques used are the method of Sub and Supersolution, Fixed point theorems and some results explored radial symmetry and some estimates for elliptic equations.
Ayed, Hela. "Analyse d'un problème d'interaction fluide-structure avec des conditions aux limites de type frottement à l'interface." Thesis, Normandie, 2017. http://www.theses.fr/2017NORMC213/document.
Повний текст джерелаThis PHD thesis is devoted to the theoretical and numerical analysis of a stationary fluid-structure interaction problem between an incompressible viscous Newtonian fluid, modeled by the 2D Stokes equations, and a deformable structure modeled by the 1D beam equations.The fluid and structure are coupled via a friction boundary condition at the fluid-structure interface.In the theoretical study, we prove the existence of a unique weak solution, under small displacements, of the fluid-structure interaction problem under a slip boundary condition of friction type (SBCF) by using Schauder fixed point theorem.In the numerical analysis, we first study a mixed finite element approximation of the Stokes equations under SBCF.We also prove an optimal a priori error estimate for regular data and we provide numerical examples.Finally, we present a fixed point algorithm for numerical simulation of the coupled problem under nonlinear boundary conditions
ARAÚJO, Damião Júnio Gonçalves. "Análise matemática de Modelos de Campo de Fase para solidificação." Universidade Federal de Campina Grande, 2008. http://dspace.sti.ufcg.edu.br:8080/jspui/handle/riufcg/1202.
Повний текст джерелаMade available in DSpace on 2018-07-19T14:01:17Z (GMT). No. of bitstreams: 1 DAMIÃO JÚNIO GONÇALVES ARAÚJO - DISSERTAÇÃO PPGMAT 2008..pdf: 506877 bytes, checksum: 8b058ceadbb68cd8c6d372656749744e (MD5) Previous issue date: 2008-04
Capes
Neste trabalho são estudados dois sistemas de equações diferenciais parciais parabólicas sujeitas a condições iniciais e de contorno. O primeiro sistema tratado representa um modelo de solidificação envolvendo uma função campo de fase. O segundo problema tratado é uma simplificação de um modelo com duas funções campo de fase para solidificação de ligas. São estudados resultados sobre existência (via Método de Ponto Fixo), regularidade, continuidade em relação aos dados iniciais e ao termo forçante e unicidade de solução dos sistemas citados.
In this work we study two parabolic partial differential equations systems subject to initial and boundary conditions. The first system treated here represents a model for solidification with a phase field function. The second system is a simplification of a two-phase field model for alloy solidification. We study results concerning existence (by FixedPoint Method), regularityand uniquenessof solution for mentioned systems.
SANTOS, Joselma Soares dos. "Soluções de sistemas de equações diferenciais elípticas via Teoria de ponto fixo em cones." Universidade Federal de Campina Grande, 2007. http://dspace.sti.ufcg.edu.br:8080/jspui/handle/riufcg/1183.
Повний текст джерелаMade available in DSpace on 2018-07-16T19:36:43Z (GMT). No. of bitstreams: 1 JOSELMA SOARES DOS SANTOS - DISSERTAÇÃO PPGMAT 2007..pdf: 482798 bytes, checksum: c569721d7def4ccf67efe94c085198f8 (MD5) Previous issue date: 2007-04
Neste trabalho usaremos a Teoria do Ponto fixo em Cones para provar a existência e multiplicidade de solução positiva radial para sistemas de equações diferenciais parciais elípticas de segunda ordem onde 0 < r1 < r2 e a,b são parâmetros não-negativos. * (O resumo original da dissertação aprenta um sistema de equação que não foi possível adiciona-lo aqui. Recomendamos o download do arquivo para acessoao resumo completo)
In this work we will use the Theory of the Fixed Point in Cones to prove the existence and multiplicity of positive solutions for systems of second-ordem elliptic differential equations where 0 < r1 < r2 and a,b are non-negative parameters. * (The original abstract of the dissertation presents an equation system that could not be added here. We recommend downloading the file for access to the full summary)
Oussaily, Aya. "Étude théorique et numérique des systèmes modélisant la dynamique des densités des dislocations." Thesis, Compiègne, 2021. https://bibliotheque.utc.fr/Default/doc/SYRACUSE/2021COMP2634.
Повний текст джерелаIn this thesis, we are interested in the theoretical and numerical studies of dislocations densities. Dislocations are linear defects that move in crystals when those are subjected to exterior stress. More generally, the dynamics of dislocations densities are described by a system of transport equations where the velocity field depends non locally on the dislocations densities. First, we are interested in the study of a one dimensional submodel of a (2 × 2) Hamilton-Jacobi system introduced by Groma and Balogh in 1999, proposed in the two dimensional case. For this system, we prove global existence and uniqueness results. Adding to that, considering nondecreasing initial data, we study this problem numerically by proposing a finite difference implicit scheme for which we show the convergence. Then, inspired by the first work, we show a more general theory which allows us to get similar results of existence and uniqueness of solution in the case of one dimensional eikonal systems. By considering nondecreasing initial data, we study this problem numerically. Under certain conditions on the velocity, we propose a finite difference implicit scheme allowing us to calculate the discrete solution and simulate then the dislocations dynamics via this model
Laurent-Brouty, Nicolas. "Modélisation du trafic sur des réseaux routiers urbains à l’aide des lois de conservation hyperboliques." Thesis, Université Côte d'Azur (ComUE), 2019. http://www.theses.fr/2019AZUR4056.
Повний текст джерелаThis thesis is devoted to the modeling of traffic flow using hyperbolic conservation laws, with a specific focus on urban applications. Urban areas are today facing severe episodes of air pollution and increasing congestion due to traffic. The objective is to overcome some of the current limitations of macroscopic traffic flow models in urban situations. We first study the seminal Aw-Rascle-Zhang model with relaxation. We prove well-posedness of the model using wave-front tracking approximations and splitting technique in a Lagrangian setting. Besides, we provide an estimate on the decay of positive waves. We then show that the solutions of the Aw-Rascle-Zhang system with relaxation converge to a weak solution of the LWR model when the relaxation parameter goes to zero. Finally, we propose a discussion on the entropy aspect of this weak solution of the LWR model. We then propose a new macroscopic traffic flow model accounting for the boundedness of traffic acceleration, which is required for physical realism. Our model is built on the coupling between the scalar conservation law accounting for the conservation of vehicles and a number of ordinary differential equations describing the trajectories of accelerating vehicles, which we treat as moving constraints. We detail a wave-front tracking algorithm to construct approximate solutions of the model, with general flux functions and show existence of solutions to the Cauchy problem for a piecewise constant initial datum. Finally, we provide numerical simulations of the model in different urban situations, from a single Riemann problem to sequences of traffic lights, and confront the results to numerical simulations of the LWR model. Finally, we introduce a new macroscopic traffic flow model with buffers on road networks. This model features buffers of finite size, enabling backward propagation of congestion on the network, and time-dependent routing functions at the junctions. The dynamics are first defined on the level of conservation laws, and then transformed in an Hamilton-Jacobi formulation. We prove existence, uniqueness and stability of the solutions with respect to the routing ratios and initial datum using a fixed-point problem in a proper Banach space. Thanks to stability, the model provides a controllable framework, using routing ratios as control parameters. This represents an advance towards solving the Dynamic Traffic Assignment (DTA) problem. In the end we detail how this framework applies to a classical road network with several intersections and finite-length links
Stötzner, Ailyn. "Optimal Control of Thermoviscoplasticity." Universitätsverlag der Technischen Universität Chemnitz, 2018. https://monarch.qucosa.de/id/qucosa%3A31887.
Повний текст джерелаDiese Arbeit ist der Untersuchung von Optimalsteuerproblemen gewidmet, denen ein quasistatisches, thermoviskoplastisches Model mit kleinen Deformationen, mit linearem kinematischen Hardening, von Mises Fließbedingung und gemischten Randbedingungen zu Grunde liegt. Mathematisch werden thermoviskoplastische Systeme durch nichtlineare partielle Differentialgleichungen und eine variationelle Ungleichung der zweiten Art beschrieben, um die elastischen, plastischen und thermischen Effekte abzubilden. Durch die Miteinbeziehung thermischer Effekte, treten verschiedene mathematische Schwierigkeiten während der Analysis des thermoviskoplastischen Systems auf, die ihren Ursprung hauptsächlich in der schlechten Regularität der nichtlinearen Terme auf der rechten Seite der Wärmeleitungsgleichung haben. Eines unserer Hauptresultate ist die Existenz einer eindeutigen schwachen Lösung, welches wir mit Hilfe von einem Fixpunktargument und unter Anwendung von maximaler parabolischer Regularitätstheorie beweisen. Zudem definieren wir die entsprechende Steuerungs-Zustands-Abbildung und untersuchen Eigenschaften dieser Abbildung wie die Beschränktheit, schwache Stetigkeit und lokale Lipschitz Stetigkeit. Ein weiteres wichtiges Resultat ist, dass die Abbildung Hadamard differenzierbar ist; Hauptbestandteil des Beweises ist die Umformulierung der variationellen Ungleichung, der sogenannten viskoplastischen Fließregel, als eine Banachraum-wertige gewöhnliche Differentialgleichung mit nichtdifferenzierbarer rechter Seite. Schließlich runden wir diese Arbeit mit numerischen Beispielen ab.
Xie, Rongzheng. "A population approach to systems of Izhikevich neurons: can neuron interaction cause bursting?" Thesis, 2020. http://hdl.handle.net/1828/11700.
Повний текст джерелаGraduate
Zelina, Michael. "Věty o pevném bodě v teorii diferenciálních rovnic." Master's thesis, 2020. http://www.nusl.cz/ntk/nusl-415504.
Повний текст джерелаZuo, Lihua. "Inverse Problems for Fractional Diffusion Equations." Thesis, 2013. http://hdl.handle.net/1969.1/151079.
Повний текст джерелаMałogrosz, Marcin. "Mathematical analysis of morphogen transport models." Doctoral thesis, 2015.
Знайти повний текст джерелаW rozprawie zostały poddane matematycznej analizie dwa modele transportu morfogenu. Oba modele bazują na koncepcji pozycyjnego sygnałowania, wprowadzonej w latach sześćdziesiątych przez Wolperta. Pozycyjne sygnałowanie tłumaczy mechanizm różnicowania się komórek i tworzenia wzorców w rozwijających się zarodkach za pomocą formowania się gradientu stężenia morfogenu. Rozważane w pracy modele to układy parabolicznych, półliniowych równań różniczkowych cząstkowych sprzężonych z równaniami zwyczajnymi. Pierwszy z analizowanych modeli, autorstwa Hufnagela i wpółpracowników, opisuje przestrzenne rozmieszczenie morfogenów i innych związków chemicznych w prostokątnej dziedzinie reprezentującej fragment tkanki. Główna matematyczna trudność w analizie modelu wynika z obecności delty Diraca w warunku brzegowym dla równania parabolicznego opisującego czasową ewolucję stężenia morfogenu. Oprócz pokazania, że rozważany układ jest dobrze postawiony oraz dowodu istnienia jedynego stanu stacjonarnego dokonana została redukcja wymiaru. Jest to ścisłe uzasadnienie tego, że model uproszczony, określony na jednowymiarowej dziedzinie, jest zredukowaną wersją pełnego modelu. Dla drugiego modelu, autorstwa Landera i współpracowników, zostały uogólnione na dziedziny dowolnego wymiaru, rezultaty otrzymane wcześniej przez Krzyżanowskiego i współpracowników. Ponadto została poprawiona topologia zbieżności rozwiązania do stanu stacjonarnego.